Your small window unit will take more than even 800 watts to start the compressor, so you'd need at least enough battery to handle starting the compressor. Even most inverters struggle with motor loads like air conditioners even if they have enough voltage on the battery side.
To get an idea of what a startup load is like, listen to this random video of a small 2-stroke generator loading up when it's trying to start the compressor of this air conditioner. You can hear the momentary deep motor startup load before it goes back to a more comfortable 500 watts or so.
I would suggest doing things in this order.
1. Make sure you have an inverter that can handle the startup load of your specific air conditioner.
2. Configure a battery that can handle the startup load without the voltage sagging below the cutoff voltage without solar connected.
3. Connect the solar.
This way you have a system where you've got a battery just large enough to kick over the air conditioner that won't cut out with every cloud that floats over.
A good 800 watt solar panel that's worth it's rating would make 500 watts on the sunniest of days for about 5 hours. Add clouds and you've got less than that. Also keep in mind that you probably won't need the air conditioner at 11am when there is just enough sun to start the air conditioner and when 4 or 5pm hits and you don't have enough sunlight to produce more power than the air conditioner needs, you'll probably want to have some storage to keep it running because that's when things really start getting hot indoors during the air conditioning season.
For what it's worth my air conditioner in my house doesn't even start for the first time of the day until 3 or 4pm usually at the end of May. In hotter weather it runs deeper cycles later in the day, there's some thermal lag involved.